The matrix equation XA+AXT=0 was recently introduced by De Terán and Dopico (2011) 3 to study the dimension of congruence orbits. They reduced the study of this equation to a number of special cases, ...several of which have not been explicitly solved. In this note we obtain an explicit, closed-form solution in the difficult Type 0–I interaction case.
We investigate paths in Bernoulli's triangles, and derive several relations linking the partial sums of binomial coefficients to the Fibonacci numbers.
En parcourant des chemins dans les triangles de ...Bernoulli, nous proposons des relations entre les sommes partielles de coefficients du binôme et les nombres de Fibonacci.
On the matrix equation XA+AXT=0 Garcia, Stephan Ramon; Shoemaker, Amy L.
Linear algebra and its applications,
03/2013, Letnik:
438, Številka:
6
Journal Article
Recenzirano
Odprti dostop
The matrix equation XA+AXT=0, which has relevance to the study of Lie algebras, was recently studied by De Terán and Dopico (Linear Algebra Appl. 434 (2011), 44–67). They reduced the study of this ...equation to several special cases and produced explicit solutions in most instances. In this note we obtain an explicit solution in one of the difficult cases, for which only the dimension of the solution space and an algorithm to find a basis of this space were known previously.
This work describes consisting of zeroes and ones mathematical model, binary matrix obtained by the arithmetical and combinatorial transformations of Pascal's triangle. Some options of a method of ...building of binary matrices by the choice of certain generatrix are listed. The example of the binary matrix formed in conditions when dimension of a matrix exceeds template length is given. The known method of creation of a binary matrix by reduction of a triangle of Pascal on the simple or compound module is given. Its comparison with the method offered in this work is carried out and the difference in creation of bigger number of fractal structures is specified. Fractal, algebraic and combinatory properties, features and distinctions of two creation of binary matrixes by means of templates 1 0 1 and 0 1 1 are described. The self-similarity properties of the binary matrixes are being examined. The theorem of the sequence of not repeating lines of the described binary matrixes is formulated and proved. The objects and their properties investigated in this work are used at the solution of tasks of the theory of information and used as models of natural processes which show property of self-organization.
Spontaneous self-assemblies of biomolecules can generate geometrical patterns. Our findings provide an insight into the mechanism of self-assembled ring pattern generation by human serum albumin ...(HSA). The self-assembly is a process guided by kinetic and thermodynamic parameters. The generated protein ring patterns display a behavior which is geometrically related to a
-simplex model and is explained through thermodynamics and chemical kinetics.
This paper, resulting from two summer programs of Research Experience for Undergraduates, examines the congruence classes of binomial coefficients to a prime square modulus as given by a fractal ...generation process for lattice path counts. The process depends on the isomorphism of partial semigroup structures associated with each iteration. We also consider integrality properties of certain critical coefficients that arise in the generation process. Generalizing the application of these coefficients to arbitrary arguments, instead of just to the prime arguments appearing in their original function, it transpires that integrality of the coefficients is indicative of the primality of the argument.
BOOL(N) Al-Haj Baddar, Sherenaz W.; Batcher, Kenneth E.
Designing Sorting Networks,
11/2011
Book Chapter
Let p be a positive integer and let N = 2p. Here we describe a certain poset , which we call BOOL (N). Let us use p-1 steps to build BOOL(N/2) out of the first N/2 keys, K0 through KN/2-1 and ...BOOL(N/2) out of the last N/2 keys KN/2 through KN-1. After that, let us use one step of N/2 comparators to compare Ki with Ki + N/2 for i = 0, 1,…, N/2-1. We conjecture that this method of building a poset of N = 2p in p steps minimizes the number of 0/1-cases . Here we also showed some ideas on starting a sorting network for N keys when N < 2p. In general, for any BOOL (N) where N> 8, one can build a poset of N-2 keys by removing keys K0 and KN-1 and adding a comparator in each step to compare the two keys that aren’t compared with any other key in that step. The poset obtained will look like BOOL(N) except with keys K0 and KN-1 removed and p extra coverings added between the keys in the top rank and keys in the bottom rank. These extra coverings eliminate strangers that are far out of place.
Electroencephalogram (EEG)-based major depressive disorder (MDD) machine learning detection models can objectively differentiate MDD from healthy controls but are limited by high complexities or low ...accuracies. This work presents a self-organized computationally lightweight handcrafted classification model for accurate MDD detection using a reference subject-based validation strategy. We used the public Multimodal Open Dataset for Mental Disorder Analysis (MODMA) comprising 128-channel EEG signals from 24 MDD and 29 healthy control (HC) subjects. The input EEG was decomposed using multilevel discrete wavelet transform with Daubechies 4 mother wavelet function into eight low- and high-level wavelet bands. We used a novel Twin Pascal’s Triangles Lattice Pattern(TPTLP) comprising an array of 25 values to extract local textural features from the raw EEG signal and subbands. For each overlapping signal block of length 25, two walking paths that traced the maximum and minimum L1-norm distances from v1 to v25 of the TPTLP were dynamically generated to extract features. Forty statistical features were also extracted in parallel per run. We employed neighborhood component analysis for feature selection, a k-nearest neighbor classifier to obtain 128 channel-wise prediction vectors, iterative hard majority voting to generate 126 voted vectors, and a greedy algorithm to determine the best overall model result. Our generated model attained the best channel-wise and overall model accuracies. The generated system attained an accuracy of 76.08% (for Channel 1) and 83.96% (voted from the top 13 channels) using leave-one-subject-out(LOSO) cross-validation (CV) and 100% using 10-fold CV strategies, which outperformed other published models developed using same (MODMA) dataset.
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